84 Worked Examples and reveals the complicated structure of the dynamics. This structure is referred to as the Lorenz attractor since it attracts orbits from all possible initial conditions.
Because of its peculiar properties the attractor has gathered considerable attention from mathematicians. For example it has been proven that the Lorenz attractor is fractal.15The model also exhibits the desired sensitivity to initial conditions and has become a canonical example of chaotic systems.
We started out with weather and wind, and via convection ended up with the relatively simple Lorenz model that exhibits complicated and even fractal dynamics.
The model does not have any predictive power, but instead makes it possible to explain a specific property of the weather, its sometimes unpredictable behaviour.
Because of its properties the model is located at the outer edge of the predictive- explanatory spectrum of models: it cannot help us predict the weather, but instead gives insight into its basic dynamics.
temperatures vary from 3000 to 30,000 K and the classes are denoted (in decreasing order of temperature): O, B, A, F, G, K, M. Each class is subdivided into ten subclasses with the help of the numbers from 0 to 9. In this classification the sun is a G2-star with a surface temperature of 5800 K. In addition it had become clear that stars vary greatly in their brightness or magnitude. In one way this is obvious since they are located at different distances from the solar system and therefore appear more or less bright on the night sky. But what we mean is not the apparent magnitude, but the absolute magnitude of a star. This is defined as the apparent magnitude a star would have if it was placed at a distance of 10 parsecs from the Earth.17 It might seem paradoxical that the absolute magnitude of a distant star can be determined, but to help them astronomers had a number of overlapping methods that allowed for accurate calculations of magnitude.
Spectral class and magnitude were essentially the only properties of stars that astronomers could observe and make use of when formulating theories of the com- position and evolution of stars. An important step towards such theories was to investigate the relationship between these two properties. In other words, how do spectral class and magnitude relate to each other?
It is precisely this relation that the HR-diagram illustrates. In the two-dimensional diagram each star is placed according to its spectral class (on the x-axis) and absolute magnitude (on the y-axis) and what emerges is an obvious pattern (see Fig.13). The dots in the diagram that correspond to stars are not randomly distributed, but most of them fall on a diagonal band, called the “main sequence”. Above this band we find a collection of stars known as “red giants” and “super giants”, and at the bottom left we find the “white dwarves”.
This pattern was a significant discovery in itself, but the most important conse- quence of the HR-diagram was that astronomers received a powerful tool for explor- ing stellar evolution. If one follows a star through its history it can be traced as a curve or orbit within the HR-diagram. Hence astronomers now had a way to illus- trate the problem and reason about it. An early example of this was Russell’s own hypothesis of how stars live and die.18He thought that a star is formed through the gravitational contraction of a cloud of gas, and that this leads to an increased tem- perature which makes the gas glow. Further contraction makes the gas even hotter and this corresponds to a movement to the left in the HR-diagram. When the star reaches the main sequence it starts to cool, but keeps on contracting. Since its size is reduced, the magnitude is lowered, and coupled to the falling temperature this leads to a movement downwards along the main sequence (see the dotted line in Fig.13).
This hypothesis was put forward in 1914, before it was known that it’s nuclear reaction in the interior of stars that provides the stellar energy, and was instead based on the assumption that the emitted energy of a star is derived from the potential energy of the collapsing gas. According to the current view of stellar evolution the
171 parsec (≈3.26 light years) is the distance at which a star exhibits a yearly angular displacement or parallax of one second of arc.
18For a detailed account of this and other examples we refer to: Tassoul, J.-L. and Tassoul, M.
(2004).A concise history of solar and stellar physics. Princeton University Press.
86 Worked Examples
Temperature (K)
Spectral class
Absolute magnitude
White dwarves
Main sequence
Red giants Red supergiants
Fig. 13 The Hertzsprung–Russel diagram that shows the relation between the brightness or absolute magnitude and the temperature or spectral class. Most stars appear on the so called “main sequence”, above it we find the giant stars and below the white dwarves. Thedotted linecorresponds to Russell’s initial hypothesis about how stars move in the diagram when they age, while thesolid lineshows the current view of stellar evolution (see the text for details)
journey of a star in the HR-diagram looks slightly different, and can be summarised as follows: A cloud of gas slowly contracts and its temperature is increased; when a critical temperature is reached nuclear reactions are initiated in the interior of the star.
This corresponds, just as in Russell’s hypothesis, to a movement to the left into the main sequence. The star spends most of its time on the main sequence, its horizontal position being determined by the size of the original gas cloud, and produces energy through the fusion of hydrogen into helium. When the hydrogen in the core is nearly exhausted helium becomes the fuel of the nuclear reactions, and this leads to a slow swelling of the star, which takes it along the “red giant branch”. This expansion eventually stops and a movement along the “horizontal branch” ensues. At this stage the star is almost exhausted of nuclear fuel and the stellar wind reduces the mass of the star. The outer parts are thrown off, and at the same time the core contracts and forms a small, white dwarf star of high density and high surface temperature that is located at the bottom left corner of the HR-diagram.
The HR-diagram initially served as way to organise and illustrate collected astro- nomical data, but turned into an important tool for reasoning about the composition and evolution of stars. This is not a formal or symbolic model that provides exact answers, but a conceptual model that aided astronomers in speculating and develop- ing theories about the birth and death of stars.